In this talk, we will discuss how numerical simulations can be used to analyze fundamental properties of the incompressible Navier–Stokes equations. The best local well-posedness results for initial value problem are obtained by perturbation in some critical spaces. We will show numerically that this problem is ill-posed outside the perturbation regime. More precisely, we construct with the help of numerical simulations two different Leray solutions having the same initial data in borderline critical spaces. We will also discuss how such ideas might be used to construct finite-time singularities.

This is joint work with Vladimír Šverák. This work was partially supported by SNSF grant 171500 and by NSF grant DMS 1362467.